The probability that a number will come up on any one throw is 1/17. The probability it will NOT come up is 16/17. The probability it will come up on the first 850 tosses but not on any of the 12500- 850= 11650 tosses is [itex](1/17)^{850}(16/17)^{11650}[/itex].
Now we can rearrange 850 "S" (for "success" in throwing that particular number) and 11650 "F" (for "failure" in throwing that particular number)
[tex]\left(\begin{array}{c}12500 \\ 850\end{array}\right)= \frac{12500!}{850!11650!}[/tex]
which can also be written 12500C850.

So the probability of rolling exactly 850 of a specific number is
[tex]_{12500}C_{850}\left(\frac{1}{17}\right)^{850}\left(\frac{16}{17}\right)^{11650}[/tex]